30 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS
17. The symbolical equations (40), (41) contain the solution of the problem, the motion of the sphere being defined by the symbolical equation (39). If we wish to exhibit the actual results by means of real quantities alone, we have only to put the right-hand members of equations (39), (40), (41) under the form U+J — 1V, and reject the imaginary part. Putting for shortness
we have m = v (1 + J~^I), and we obtain
........................... (43),
n
^ 3 / ^\. zvaj 2va\ vaj
3 F
~ e-v{~r-°)\ co$(nt
+ — )sia(7rf
3 \ p = - i pacn 1 '+ j sin nt
i ... ......... (44),
va
(45).
v ;
The reader will remark that the %, ^, p of the present article are not the same as the f , -v/r, p of the preceding. The latter are the imaginary expressions, of which the real parts constitute the former. It did not appear necessary to change the notation.
When p,' = 0, v = GO , and ^ reduces itself to
In this case we get from (26)
and Edr + ®rdd is an exact differential d*j> where